To determine the domain and range of the function [tex]\( f(x) = \left(\frac{1}{6}\right)^x + 2 \)[/tex], let's analyze the function step-by-step.
### Domain
The domain of a function is the set of all possible input values (x-values) for which the function is defined.
1. The base [tex]\(\frac{1}{6}\)[/tex] is a positive number and can be raised to any real power. There are no restrictions on the exponent [tex]\( x \)[/tex].
2. Since there are no restrictions on [tex]\( x \)[/tex], the domain of [tex]\( f(x) \)[/tex] is all real numbers.
So, the domain is:
[tex]\[ \{x \mid x \text{ is a real number}\} \][/tex]
### Range
The range of a function is the set of all possible output values (y-values).
1. The term [tex]\( \left(\frac{1}{6}\right)^x \)[/tex] is always positive since any positive number raised to any power remains positive.
2. As [tex]\( x \)[/tex] increases, [tex]\( \left(\frac{1}{6}\right)^x \)[/tex] approaches 0 but never actually reaches 0. Similarly, as [tex]\( x \)[/tex] decreases, [tex]\( \left(\frac{1}{6}\right)^x \)[/tex] grows larger.
3. Adding 2 to [tex]\( \left(\frac{1}{6}\right)^x \)[/tex], we see that the output is always greater than 2. No matter how small [tex]\( \left(\frac{1}{6}\right)^x \)[/tex] gets (even if it approaches 0), [tex]\( f(x) \)[/tex] will always be slightly greater than 2.
Therefore, the range of [tex]\( f(x) \)[/tex] is:
[tex]\[ \{y \mid y > 2\} \][/tex]
### Final Answer
Based on the above reasoning, the correct domain and range for the function [tex]\( f(x) = \left(\frac{1}{6}\right)^x + 2 \)[/tex] are:
- Domain: [tex]\(\{ x \mid x \text{ is a real number} \}\)[/tex]
- Range: [tex]\(\{ y \mid y > 2 \}\)[/tex]
Thus, the correct choice from the given options is the third one:
[tex]\[ \text{domain: } \{ x \mid x \text{ is a real number} \}; \text{ range: } \{ y \mid y > 2 \} \][/tex]
So, the answer is:
3
